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2 votes
0 answers
254 views

Forgetting extra structure inducing Symmetries

This is a major edit of the original post after receiving helpful comments. It is often the case when one adds additional structure to make a problem more tractable. When one attempts to forget this ...
3 votes
1 answer
1k views

Maximizing Sparsity in l1 Minimization?

Consider the optimization problem $$\min_x ||Ax||_1 + \lambda||x-b||^2,$$ where $A \in \mathbb{R}^{n \times n}$, $x,b \in \mathbb{R}^n$ and $\lambda$ is strictly greater than 0. (This problem is ...
7 votes
1 answer
2k views

Why is 3 a bad constant in the Vitali covering lemma?

Hi, recently I had to do with the Hardy-Littlewood maximal function and we used there the Vitali covering lemma with constant 5. Then, given an advice, I proved it with constant k>3. But I cannot ...
11 votes
2 answers
587 views

Algebraicity of the completion of a field? Finiteness?

At the end of my 8410 class today (see http://alpha.math.uga.edu/~pete/MATH8410.html if you care), one of my students asked me the following very interesting question: Let $(K,|\ |)$ be a normed field,...
9 votes
1 answer
604 views

Which changes of metric fix all open balls of a metric space?

In an earlier question, I was interested in counting the number of metric spaces on N points, where I considered two metric spaces to be the same if they had the same collection of open balls. Two ...
26 votes
4 answers
4k views

What is the "right" universal property of the completion of a metric space?

I'm a little embarrassed to ask this one, but it could help for a class I'm teaching, so here goes: Let $X$ be a metric space. We all know that $X$ admits a completion, which is a complete metric ...
6 votes
2 answers
365 views

Why is GL(n,C)/U(n) a CAT(0) space?

The title says it all. In one of his answers to the question "Convex hull in CAT(0)" (I don't have the points to post a link, if someone doesn't mind link-ifying this that would be cool), Greg ...
6 votes
2 answers
6k views

Minimum-area bounding quadrilateral algorithm

There are a few algorithms around for finding the minimal bounding rectangle (OBB) containing a given (convex) polygon. Does anybody know about an algorithm for finding a minimal-area bounding ...
4 votes
3 answers
852 views

Variational characterization of curvature?

Consider a surface $S$ smoothly embedded in $\mathbb{R}^3$. Classically, the (Riemannian) curvature of $S$ is described by the second fundamental form, which is constructed from partial derivatives ...
7 votes
2 answers
726 views

Zeta function for curves in a manifold

Motivation In the analogy between prime numbers and knots, the prime number is thought sometimes as the circle of length $l([p]) = \text{log}\,p$. This is so you can express the zeta function as $$ \...
12 votes
3 answers
1k views

distance regular metric spaces

A metric space (V,d) will be called distance regular if for every distances a>0, b, c a nonnegative integer p(a,b,c) is defined, so that whenever d(B,C)=a, there are precisely p(a,b,c) points A ...
4 votes
2 answers
818 views

Number of independent distances between n points in d-dimensional Euclidean space?

There are $\binom{n}{2}$ distances between $n$ points in $\mathbb{R}^d$. Not all of them can be chosen freely if $n$ exceeds the number $n_d = d + 1$. If $n = n_d$ we obviously have $\binom{d+1}{2}$ ...
7 votes
2 answers
366 views

Simplicial and cubical decompositions of low valence

Every surface can be triangulated in such a way that at most 7 trianlges meet at one vertex. Every surface can be decomposed in squares such that at every vertex at most 5 suqares meet. For surfaces ...
1 vote
2 answers
6k views

If a quadratic form is positive definite on a convex set, is it convex on that set?

Consider a real symmetric matrix $A\in\mathbb{R}^{n \times n}$. The associated quadratic form $x^T A x$ is a convex function on all of $\mathbb{R}^n$ iff $A$ is positive semidefinite, i.e., if $x^T A ...
8 votes
6 answers
1k views

Combinatorial distance ≡ Euclidean distance

Definition: A polytope has property X iff there is a function f:N+ → R+ such that for each pair of vertices vi, vj the following holds: disteuclidean(vi, vj) = f(distcombinatorial(vi, vj)) with ...
11 votes
4 answers
958 views

Geometry of the multilagrangian Grassmannian

Let's introduce the following variety $MG(3,6)$, which is a "multisymplectic" analog of a Lagrangian Grassmannian $LG(3,6)$. Consider a 3-form $\omega = dx1 \wedge dx2 \wedge dx^3 - dx4 \wedge dx5 \...
10 votes
1 answer
835 views

what was Hilbert's geometric construction in his 17th problem?

Hilbert's 17th problem asked if a nonnegative real polynomial is the sum of squares of rational functions. It was answered affirmative by Artin in around 1920. However, in his speech, he also asked if ...
-4 votes
4 answers
677 views

What is the max number of points in R^3, interconnected by generic curves?

The largest complete graph that embeds in 2 dimensions is $K_4$, while the largest complete graph that embeds in 3 dimensions is $K_{\infty}$, right? However, I don't know any constructive proof of it....
11 votes
2 answers
2k views

Characterization of Riemannian metrics

This is probably an insanely hard question, but given an abstract metric space, is there some way to determine whether it's a manifold with a Riemannian, or more generally a Finslerian, metric? If ...
1 vote
1 answer
335 views

Systems of conics

It seems well-known that the system of conics given by $\frac{x^2}{a^2}+\frac{y^2}{a^2-c^2}=1$ for $c>0$ fixed and $a \in (0,c)\cup(c,\infty)$ varying is orthogonal: whenever two of these curves ...
17 votes
1 answer
1k views

Tropical mathematics and enriched category theory

Is there a connection between tropical mathematics and the Lawvere enriched category theory approach to metric spaces? I guess I will give a partial answer to this below, but I mean can they be ...
20 votes
4 answers
2k views

Minkowski sum of small connected sets

Suppose that the convex hull of the Minkowski sum of several compact connected sets in $\mathbb R^d$ contains the unit ball centered at the origin and the diameter of each set is less than $\delta$. ...
13 votes
7 answers
2k views

Upper bound on the area of a midpoint pentagon?

Starting with a convex pentagon P, we define the "middle polygon" Q, whose vertices are the middle points of the sides of the initial pentagon. The ratio between the areas of this polygons seem to ...
14 votes
6 answers
1k views

How to smootly interpolate between möbius transformations?

If you have two Möbius transformations represented as: $f(z) = \frac{az + b}{cz + d}$ $g(z) = \frac{pz + q}{rz + s}$ where $a, b, c, d, p, q, r, s, z \in \mathbb{C}$ Is it possible to derive a ...
7 votes
2 answers
1k views

What are the possible images of a square under an area-preserving map?

Let S be the open unit square in R^2: the set of points (x,y) with 0 < x < 1 and 0 < y < 1. Consider an area-preserving smooth map S --> R^2, that is, a map whose Jacobian has determinant ...
3 votes
3 answers
311 views

Are there infinite sets of stellations of polyhedra?

Lists of stellations of polyhedrons are given particular rules like in the book The Fifty Nine Icosahedra which follows "Miller's Rules". There seems to be no "correct" ruleset to use, so more ...
12 votes
3 answers
530 views

Making an l_2 distance out of l_1 distance

If we think of the l1 distance as a grid-distance between points, then we can think of l2 distance as what we get when we "shortcut" the grid by going "inside" a cell. Making the grid finer doesn't ...

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